For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusic conjectured that for every fixed $j\geq 2$, the limit $$\lim_{n\to\infty} s_j(n)/n$$ does not exist. In this paper, we prove this conjecture in the affirmative. Moreover, we define a new class of iterated remainder sets $T_j(n)$ that naturally arises from the study of the length of the Engel series expansion of a rational number. We analogously study the asymptotic behavior of $t_j(n) := |T_j(n)|$. We show that $$\lim_{n\to\infty}\frac{t_j(n)}n$$ exists precisely for $j\in\{0,1\}$ and fails to exist for every fixed integer $j\geq2$.
Let $[a_1(x),a_2(x),\ldots,a_n(x),\ldots]$ be the continued fraction expansion of irrational $x\in[0,1)$, and let $q_n(x)$ be the denominator of the $n$-th convergent. In this paper, we study how the growth rate of $a_{n+1}(x)$ on a prescribed logarithmic size interacts with its upper growth rate relative to $q_n(x)$....
We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality $q\ge2$, we prove $\ma...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\...
For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. A strictly increasing sequence of integers $x_1<\cdots<x_k$ is an \emph{ascending wave} if its consecutive differences are nondecreasing. Let $g(n)$ be the largest integer $k$ such that every set $A\subseteq[n]$ with $|A|\ge n/2$ contains an ascending wave of leng...
Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n...
T. Minamide, Haruka Sakai, Y. Tanigawa· 1 citation
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